Theorem cleqh

Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions as in dfcleq . See also cleqf . (Contributed by NM, 26-May-1993) (Proof shortened by Wolf Lammen, 14-Nov-2019) Remove dependency on ax-13 . (Revised by BJ, 30-Nov-2020)

	Ref	Expression
Hypotheses	cleqh.1	$$⊢y\in A\to \forall xy\in A$$
	cleqh.2	$$⊢y\in B\to \forall xy\in B$$
Assertion	cleqh	$$⊢A=B\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		cleqh.1	$$⊢y\in A\to \forall xy\in A$$
2		cleqh.2	$$⊢y\in B\to \forall xy\in B$$
3		dfcleq	$$⊢A=B\leftrightarrow \forall y\left(y\in A\leftrightarrow y\in B\right)$$
4		nfv	$$⊢Ⅎy\left(x\in A\leftrightarrow x\in B\right)$$
5	1	nf5i	$$⊢Ⅎxy\in A$$
6	2	nf5i	$$⊢Ⅎxy\in B$$
7	5 6	nfbi	$$⊢Ⅎx\left(y\in A\leftrightarrow y\in B\right)$$
8		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
9		eleq1w	$$⊢x=y\to \left(x\in B\leftrightarrow y\in B\right)$$
10	8 9	bibi12d	$$⊢x=y\to \left(\left(x\in A\leftrightarrow x\in B\right)\leftrightarrow \left(y\in A\leftrightarrow y\in B\right)\right)$$
11	4 7 10	cbvalv1	$$⊢\forall x\left(x\in A\leftrightarrow x\in B\right)\leftrightarrow \forall y\left(y\in A\leftrightarrow y\in B\right)$$
12	3 11	bitr4i	$$⊢A=B\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in B\right)$$